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\title{\$SPAD/src/input repa6.input}
\author{The Axiom Team}
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-- This file demonstrates Representation Theory in Scratchpad
-- using the packages REP1, REP2, IRSN and SGCF, which are the
-- abbreviations for RepresentationPackage1, RepresentationPackage2
-- IrrRepSymNatPackage and SymmetricGroupCombinatoricFunctions.
 
-- authors: Holger Gollan, Johannes Grabmeier
-- release 1.0  09/30/87
-- release 2.0  11/10/88: J. Grabmeier: add functions from IRSN
-- release 2.1  08/04/89: J. Grabmeier: adjusting to new PERM
--   and modified REP1
-- release 2.2  06/05/89: J. Grabmeier: adjusting to new algebra
-- release 2.3  08/20/89: J. Grabmeier: minor adjustments
 
-- In the sequel we show how to get all 2-modular irreducible
-- representations of the alternating group A6.
 
-- We generate A6 by the permutations threecycle x=(1,2,3)
-- and the 5-cycle y=(2,3,4,5,6)
 
genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
 
-- pRA6 is the permutation representation over the Integers...
 
pRA6 := permutationRepresentation (genA6, 6)
 
-- ... and pRA6m2 is the permutation representation over PrimeField 2:
 
pRA6m2 : List Matrix PrimeField 2 := pRA6
 
-- Now try to split pRA6m2:
 
sp0 := meatAxe pRA6m2
 
-- We have found the trivial module as a factormodule
-- and a 5-dimensional submodule.
 
dA6d1 := sp0.2
 
-- Try to split again...
 
sp1 := meatAxe sp0.1
 
-- ... and find a 4-dimensional submodule, say dA6d4a, and the
-- trivial one again.
 
dA6d4a := sp1.2
 
-- Now we want to test, whether dA6d4a is absolutely irreducible...
 
isAbsolutelyIrreducible? dA6d4a
 
-- ...and see: dA6d4a is absolutely irreducible.
-- So we have found a second irreducible representation.
 
-- Now construct a representation from reducing an irreducible one
-- of the symmetric group S_6 over the integers taken mod 2
-- What is the degree of the representation belonging to partition
-- [2,2,1,1]?
 
 
-- lambda : PRTITION := partition [2,2,1,1]
lambda := [2,2,1,1]
dimIrrRepSym lambda
 
-- now create the restriction to A6:
 
d2211  := irrRepSymNat(lambda, genA6)
 
-- ... and d2211m2 is the representation over PrimeField 2:
 
d2211m2 : List Matrix PrimeField 2 := d2211
 
-- and split it:
 
sp2 := meatAxe d2211m2
 
-- A 5 and a 4-dimensional one.
 
-- we take the 4-dimensional one, say dA6d4b:
 
dA6d4b := sp2.1
 
-- This is absolutely irreducible, too ...
 
isAbsolutelyIrreducible? dA6d4b
 
-- ... and dA6d4a and dA6d4b are not equivalent:
 
areEquivalent? ( dA6d4a , dA6d4b )
 
-- So the third irreducible representation is found.
 
-- Now construct a new representation with the help of the tensorproduct
 
dA6d16 := tensorProduct ( dA6d4a , dA6d4b )
 
-- and try to split it...
 
sp3 := meatAxe dA6d16
 
-- The representation is irreducible, but may be not
-- absolutely irreducible.
 
isAbsolutelyIrreducible? dA6d16
 
-- So let's try the same over the field with 4 elements:
 
gf4 := FiniteField(2,2)
 
dA6d16gf4 : List Matrix gf4 := dA6d16
sp4 := meatAxe dA6d16gf4
 
-- Now we find two 8-dimensional ones, dA6d8a and dA6d8b.
 
dA6d8a : List Matrix gf4  := sp4.1
dA6d8b : List Matrix gf4  := sp4.2
 
-- Both are absolutely irreducible...
 
isAbsolutelyIrreducible? dA6d8a
isAbsolutelyIrreducible? dA6d8b
 
-- and they are not equivalent...
 
areEquivalent? ( dA6d8a, dA6d8b )
 
-- So we have found five absolutely irreducible representations of A6
-- in characteristic 2.
 
-- The theory tells us that there are no more irreducible ones.
-- Here again are all absolutely irreducible 2-modular
-- representations of A6
 
dA6d1
dA6d4a
dA6d4b
dA6d8a
dA6d8b
 
-- And here again is the irreducible, but not absolutely irreducible
-- representations of A6 over PrimeField 2
 
dA6d16
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